Author:
WILKES EDWARD D.,BASARAN OSMAN A.
Abstract
Viscous liquid drops undergoing forced oscillations have been shown to exhibit
hysteretic deformation under certain conditions both in experiments and by solution
of simplified model equations that can only provide a qualitative description of their
true response. The first hysteretic deformation results for oscillating pendant drops
obtained by solving the full transient, nonlinear Navier–Stokes system are presented
herein using a sweep procedure in which either the forcing amplitude A or frequency
Ω is first increased and then decreased over a given range. The results show the
emergence of turning-point bifurcations in the parameter space of drop deformation
versus the swept parameter. For example, when a sweep is carried out by varying Ω
while holding A fixed, the first turning point occurs at
Ω ≡ Ωu as Ω is being increased
and the second one occurs at
Ω ≡ Ωl < Ωu
as Ω is being decreased. The two turning
points shift further from each other and toward lower values of the swept parameter as
Reynolds number Re is increased. These turning points mark the ends of a hysteresis
range within which the drop may attain either of two stable steady oscillatory states –
limit cycles – as identified by two distinct solution branches. In the hysteresis range,
one solution branch, referred to as the upper solution branch, is characterized by
drops having larger maximum deformations compared to those on the other branch,
referred to as the lower solution branch. Over the range
Ωl [les ] Ω [les ] Ωu, the sweep
procedure enables detection of the upper solution branch which cannot be found if
initially static drops are set into oscillation as in previous studies of forced oscillations
of supported and captive drops, or liquid bridges. The locations of the turning points
and the associated jumps in drop response amplitudes observed at them are studied
over the parameter ranges 0.05 [les ] A [les ] 0.125, 20 [les ] Re [les ] 40,
and gravitational Bond number 0 [les ] G [les ] 1. Critical
forcing amplitudes for onset of hysteresis are also
determined for these Re values. The new findings have important ramifications in
several practical applications. First, that
Ωu − Ωl
increases as Re increases overcomes the limitation which is inherent
to the current practice of inferring the surface
tension and/or viscosity of a bridge/drop liquid from measurement of its resonance
frequencies (Chen & Tsamopoulos 1993; Mollot et al. 1993). Moreover, that the value
of A for onset of hysteresis can be as low as 5% of the drop radius, or lower, has
important implications for other free-surface flows such as coating flows.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
25 articles.
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